Published March 2, 2017
Tags: Analysis, History, Logic
Children sometimes amuse themselves searching for the biggest number. After trying millions, billions and trillions, they realize that there is no end to the game: however big a number may be, we can always add 1 to produce a bigger number: the set of counting numbers is infinite. The concept of infinity has intrigued philosophers since antiquity, and it leads to many surprises and paradoxical results [TM110 or search for “thatsmaths” at irishtimes.com].
Continue reading ‘Enigmas of Infinity’
The claim is often made that mathematical results are immutable. Once proven, they remain forever valid. But things are not so simple. There are problems at the very core of mathematics that cast a shadow of uncertainty. We can never be absolutely sure that the foundations of our subject are rock-solid [TM104 or search for “thatsmaths” at irishtimes.com].
Left: Plato and Aristotle. Centre: Pythagoras. Right: Euclid [Raphael, The School of Athens]
The ancient Greeks put geometry on a firm footing. Euclid set down a list of axioms, or basic intuitive assumptions. Upon these, the entire edifice of Euclidean geometry is constructed. This axiomatic approach has been the model for mathematics ever since.
Continue reading ‘The Shaky Foundations of Mathematics’
Published March 10, 2016
Tags: Algorithms, Logic, Music
The links between mathematics and music are manifold. Mathematics can be set to music in a simple but surprising manner. For the award ceremony of the Gödel Medal in 2014, a musical interpretation of Gödel’s incompleteness Theorems was written by Danish composer Niels Marthinsen. It encodes the basic axioms of number theory that form the focus of Gödel’s Theorems.
The Peano Axioms in symbolic form.
Continue reading ‘Peano Music’
This week’s That’s Maths column in The Irish Times (TM058, or search for “thatsmaths” at irishtimes.com) is about George Boole, the first Professor of Mathematics at Queen’s College Cork.
Continue reading ‘The Year of George Boole’
Published July 31, 2014
Tags: Logic, Set Theory
Many of us recall the sense of wonder we felt upon learning that there is no biggest number; for some of us, that wonder has never quite gone away. It is obvious that, given any counting number, one can be added to it to give a larger number. But the implication that there is no limit to this process is perplexing.
Georg Cantor (1845 – 1918) around 1870 (left) and in later life (right).
Continue reading ‘Degrees of Infinity’
Published May 9, 2013
Tags: Logic, Topology
One of the most amazing and counter-intuitive results in mathematics was proved in 1924 by two Polish mathematicians, Stefan Banach and Alfred Tarski. Banach was a mathematical prodigy, and was the founder of modern functional analysis. Tarski was a logician, educated at the University of Warsaw who, according to his biographer, “changed the face of logic in the twentieth century” through his work on model theory.
Continue reading ‘The Loaves and the Fishes’